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Confinement and the superfluid density in theories of the underdoped cuprates, and strongly commensurate dirty bosons in the large-N limit

Matthew Case

Published Jan 1, 2005• 0 citations
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Abstract

In this thesis, we study various issues arising from the QED theory of underdoped, high temperature superconductors in 2+1 dimensions. The theory breaks up roughly into two sectors: fermionic and bosonic. With regard to the fermionic sector, we consider confinement of the emergent gauge field which we take to be compact. In the absence of fermions, the interaction between monopoles is Coulombic and the well known result is that the pure gauge theory is permanently confining. With the addition of fermions, the interaction becomes logarithmic, and an analogy with the usual Kosterlitz-Thouless transition suggests a deconfinement transition for the fermions. We show, however, that, when screening is taken into account, the deconfined phase is destabilized and fermions remain permanently confined. The bosonic sector models Cooper pair phase fluctuations, whose effect on the depletion of the superfluid density we examine in two separate studies. In the first of these, we study the quantum XY model, and show that the quasi-two dimensionality, low critical temperatures and large d-wave gap characteristic of underdoped cuprates severely constrain the form of the superfluid density. Under these assumptions, we find that phase fluctuations alone are insufficient to account for recent observations of deviations from Uemura scaling, and that the quasiparticle contribution is a necessity. We use our results to satisfactorily fit the recent data. In the second study, we model the cuprates by a layered system of interacting bosons and examine the collective excitations in this system. Depending on the anisotropy and the interaction strength, we find find four different regimes of temperature dependence of the superfluid density. We argue that interactions in the underdoped cuprates are effectively short-ranged and weak. Finally, we study the related issue of disordered, interacting bosons in the large-N limit and 2 strong commensuration. Perturbatively at weak disorder and numerically at strong, we show that th screening of the random potential due to interactions is insufficient to delocalize the single-particl states so that no superfluid transition occurs from the Mott insulator.

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This Thesis analyzes the role of a mean-field theory and of the fluctuations in two different fermionic systems: the BCS-BEC crossover and high-Tc superconducting cuprates. In the first part of the Thesis we introduce the mean-field theory and the Gaussian-level fluctuations for the BCS-BEC crossover, i.e. the continuous evolution observed in ultracold neutral fermionic systems, from a weakly-coupled regime where fermions form Cooper pairs, to a strongly-coupled regime where the fermions at first form bosonic molecular dimers, subsequently undergoing Bose-Einstein condensation for low enough temperatures. We then analyze some problems related to the crossover: the condensate fraction for a spin-unbalanced Fermi gas, the Beliaev decay for collective excitations and the case of a two-dimensional Fermi gas, for which we calculate the first sound velocity and the Berezinskii-Kosterlitz-Thouless critical temperature, in very good agreement with recent experimental data. Finally we analyze a regularization procedure for the equation of state in the deep-BEC limit, allowing one to analytically derive the correct ratio between the fermionic and bosonic scattering lengths. In the second part of the Thesis we introduce a gauge approach to superconductivity in cuprates in which the hole is decomposed as the product of a spinful neutral particle, the spinon, and a spinless charged particle, the holon. The statistics of each particle is modified by binding it to a gauge field providing a statistical flux, this approach being analogous to Chern-Simons bosonization. We thus obtain a model characterized by three characteristic temperatures corresponding, respectively, to the appearance of a finite density of incoherent holon pairs, of a finite density of incoherent spinon pairs and, finally, to the phase coherence leading to superconductivity. Specifically in the present Thesis within this model we study the superfluid density, demonstrating that the spinon and holon contributions sum according to a Ioffe-Larking-like rule, analogous to that they found for resistivity, showing excellent agreement with experimental data in the moderate underdoping up to optimal doping region. We also demonstrated that the formalism reproduces the quasi-universality observed in experimental data at different dopings and allows for an analytical derivation of a relation that approximates the Uemura relation for moderate dopings.

•Feb 1, 2016•1 citation

Large-Ntheory of strongly commensurate dirty bosons: absence of a transition in two dimensions

Matthew Case, Igor F. Herbut

The spherical limit of strongly commensurate dirty bosons is studied perturbatively at weak disorder and numerically at strong disorder in two dimensions (2D). We argue that disorder is not perfectly screened by interactions and consequently that the ground state in the effective Anderson localization problem always remains localized. As a result there is only a gapped Mott insulator phase in the theory. Comparisons with other studies and the parallel with disordered fermions in 2D are discussed. We conjecture that while for the physical cases N = 2 (XY) and N = 1 (Ising) the theory should have the ordered phase, it may not for N = 3 (Heisenberg).

arXiv:cond-mat/0105442•Journal of Physics A Mathematical and General•Sep 14, 2001•4 citations

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We study the stability of the spin gap phase in the U (1) slave-boson theory of the t - J model in connection with the underdoped cuprates. We approach the spin gap phase from the superconducting state and consider the quantum phase transition of the slave bosons at zero temperature by introducing vortices in the boson superfluid. At finite temperatures, the properties of the bosons are different from those in the strange-metal phase and lead to modified gauge field fluctuations. As a result, the spin gap phase can be stabilized in the quantum-critical (QC) and quantum-disordered (QD) regime of the boson system. We also show that the regime of QD bosons with the paired fermions can be regarded as a strong-coupling version of the recently proposed nodal liquid theory.

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The phase diagrams and phase transitions of bosons with short-ranged repulsive interactions moving in periodic and/or random external potentials at zero temperature are investigated with emphasis on the superfluid-insulator transition induced by varying a parameter such as the density. Bosons in periodic potentials (e.g., on a lattice) at T=0 exhibit two types of phases: a superfluid phase and Mott insulating phases characterized by integer (or commensurate) boson densities, by the existence of a gap for particle-hole excitations, and by zero compressibility. Generically, the superfluid onset transition in d dimensions from a Mott insulator to superfluidity is ``ideal,'' or mean field in character, but at special multicritical points with particle-hole symmetry it is in the universality class of the (d+1)-dimensional XY model. In the presence of disorder, a third, ``Bose glass'' phase exists. This phase is insulating because of the localization effects of the randomness and analogous to the Fermi glass phase of interacting fermions in a strongly disordered potential.The Bose glass phase is characterized by a finite compressibility, no gap, but an infinite superfluid susceptibility. In the presence of disorder the transition to superfluidity is argued to occur only from the Bose glass phase, and never directly from the Mott insulator. This zero-temperature superfluid-insulator transition is studied via generalizations of the Josephson scaling relation for the superfluid density at the ordinary \ensuremath{\lambda} transition, highlighting the crucial role of quantum fluctuations. The transition is found to have a dynamic critical exponent z exactly equal to d and correlation length and order-parameter correlation exponents \ensuremath{\nu} and \ensuremath{\eta} which satisfy the bounds \ensuremath{\nu}\ensuremath{\ge}2/d and \ensuremath{\eta}\ensuremath{\le}2-d, respectively. It is argued that the superfluid-insulator transition in the presence of disorder may have an upper critical dimension ${d}_{c}$ which is infinite, but a perturbative renormalization-group calculation wherein the critical exponents have mean-field values for weak disorder above d=4 is also discussed. Many of these conclusions are verified by explicit calculations on a model of one-dimensional bosons in the presence of both random and periodic potentials. The general results are applied to experiments on $^{4}\mathrm{He}$ absorbed in porous media such as Vycor. Some measurable properties of the superfluid onset are predicted exactly [e.g., the exponent x relating the \ensuremath{\lambda} transition temperature to the zero-temperature superfluid density is found to be d/2(d-1)], while stringent bounds are placed on others. Analysis of preliminary data is consistent with these predictions.

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Superfluidity in Strongly Correlated Systems

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We study superfluidity in strongly correlated systems by focusing three features as follows. First, interplay between superfluidity and other order parameters near quantum critical points is studied. The onset positions of the order parameters with or without superconductivity determines how they interact with each other. For the spin density wave (SDW) transition, we find that the quantum critical point shifts by order |∆|, where ∆ is pairing amplitude, so that the region of SDW order is smaller in the superconductor than in the metal. This shift is larger than the |∆|2 shift predicted by theories of competing orders which ignore Fermi surface effects. Second, we study a doped antiferromagnetic Mott insulator motivated by copper based high temperature superconductors. Based on spin and charge fractionalization, the algebraic charge liquid phase whose bosonic spinons have energy gap but fermionic holons form Fermi surfaces and its superconductivity are studied. We show fluctuations of local SDW order orientation lead to d-wave superconductivity, and its quantum critical pairing mechanisms are investigated. Also, a theory of bound states between spinons and holons is studied. We show that the fractionalization induces a doubling electron-like quasiparticles, which naturally induces the fractionalized Fermi liquid. The bound state theory describes exotic properties of the underdoped Cuprates such as “nodal-antinodal dichotomy” at a mean field level description. At last, we study superfluidity of fermions in optical lattices near the Feshbach resonance. Due to divergence of scattering length at the resonance, universality is observed and characteristics of superfluidity has been studied. We extend such universal properties by considering superfluid-insulator transition with an even integer number of fermions per unit cell. The resulting insulator interpolates between a band insulator of fermions and a Mott insulator of fermion pairs.

•Dec 1, 2011•0 citations

Two-dimensional Bose liquid with strong gauge-field interaction

M. V. Feigel’man, V. B. Geshkenbeǐn, L. B. Ioffe +1 more

Two unrelated problems can be reduced to a model of a Bose gas interacting with a gauge field: (i) the effect of thermal fluctuations on a system of vortices in bulk superconductors in fields ${\mathit{H}}_{\mathit{c}1}$\ensuremath{\ll}H\ensuremath{\ll}${\mathit{H}}_{\mathit{c}2}$, and (ii) charged, spinless excitations in two-dimensional (2D) strongly correlated electron systems. Both problems are important for the theory of high-temperature superconductors. We study this model in three regimes: at finite temperatures, assuming that the gauge field is purely transverse; at T=0, for the purely static (2D Coulomb) interaction; and at T=0, for a weak Coulomb interaction and a strong transverse one. Transverse interactions suppress the temperature of the superfluid transition significantly. A sufficiently strong transvese interaction is shown to produce a phase separation as the temperature decreases (in the absence of Coulomb repulsion). If there is Coulomb repulsion, the ground state does not have off-diagonal long-range order but the superfluid density is not zero unless the Coulomb constant exceeds a critical value. Sufficiently strong coupling to the transverse field destroys superfluidity as well. In the normal state formed at large couplings, the translational invariance is intact. We propose a bosonic ground state that is not superfluid at T=0. We discuss the implications of these results both for vortex liquids and strongly correlated electron systems.

Physical review. B, Condensed matter•Dec 1, 1993•82 citations
In this thesis, we study various issues arising from the QED theory of underdoped, high temperature superconductors in 2+1 dimensions. The theory breaks up roughly into two sectors: fermionic and bosonic. With regard to the fermionic sector, we consider confinement of the emergent gauge field which we take to be compact. In the absence of fermions, the interaction between monopoles is Coulombic and the well known result is that the pure gauge theory is permanently confining. With the addition of fermions, the interaction becomes logarithmic, and an analogy with the usual Kosterlitz-Thouless transition suggests a deconfinement transition for the fermions. We show, however, that, when screening is taken into account, the deconfined phase is destabilized and fermions remain permanently confined. The bosonic sector models Cooper pair phase fluctuations, whose effect on the depletion of the superfluid density we examine in two separate studies. In the first of these, we study the quantum XY model, and show that the quasi-two dimensionality, low critical temperatures and large d-wave gap characteristic of underdoped cuprates severely constrain the form of the superfluid density. Under these assumptions, we find that phase fluctuations alone are insufficient to account for recent observations of deviations from Uemura scaling, and that the quasiparticle contribution is a necessity. We use our results to satisfactorily fit the recent data. In the second study, we model the cuprates by a layered system of interacting bosons and examine the collective excitations in this system. Depending on the anisotropy and the interaction strength, we find find four different regimes of temperature dependence of the superfluid density. We argue that interactions in the underdoped cuprates are effectively short-ranged and weak. Finally, we study the related issue of disordered, interacting bosons in the large-N limit and 2 strong commensuration. Perturbatively at weak disorder and numerically at strong, we show that th screening of the random potential due to interactions is insufficient to delocalize the single-particl states so that no superfluid transition occurs from the Mott insulator.
This Thesis analyzes the role of a mean-field theory and of the fluctuations in two different fermionic systems: the BCS-BEC crossover and high-Tc superconducting cuprates. In the first part of the Thesis we introduce the mean-field theory and the Gaussian-level fluctuations for the BCS-BEC crossover, i.e. the continuous evolution observed in ultracold neutral fermionic systems, from a weakly-coupled regime where fermions form Cooper pairs, to a strongly-coupled regime where the fermions at first form bosonic molecular dimers, subsequently undergoing Bose-Einstein condensation for low enough temperatures. We then analyze some problems related to the crossover: the condensate fraction for a spin-unbalanced Fermi gas, the Beliaev decay for collective excitations and the case of a two-dimensional Fermi gas, for which we calculate the first sound velocity and the Berezinskii-Kosterlitz-Thouless critical temperature, in very good agreement with recent experimental data. Finally we analyze a regularization procedure for the equation of state in the deep-BEC limit, allowing one to analytically derive the correct ratio between the fermionic and bosonic scattering lengths. In the second part of the Thesis we introduce a gauge approach to superconductivity in cuprates in which the hole is decomposed as the product of a spinful neutral particle, the spinon, and a spinless charged particle, the holon. The statistics of each particle is modified by binding it to a gauge field providing a statistical flux, this approach being analogous to Chern-Simons bosonization. We thus obtain a model characterized by three characteristic temperatures corresponding, respectively, to the appearance of a finite density of incoherent holon pairs, of a finite density of incoherent spinon pairs and, finally, to the phase coherence leading to superconductivity. Specifically in the present Thesis within this model we study the superfluid density, demonstrating that the spinon and holon contributions sum according to a Ioffe-Larking-like rule, analogous to that they found for resistivity, showing excellent agreement with experimental data in the moderate underdoping up to optimal doping region. We also demonstrated that the formalism reproduces the quasi-universality observed in experimental data at different dopings and allows for an analytical derivation of a relation that approximates the Uemura relation for moderate dopings.
The spherical limit of strongly commensurate dirty bosons is studied perturbatively at weak disorder and numerically at strong disorder in two dimensions (2D). We argue that disorder is not perfectly screened by interactions and consequently that the ground state in the effective Anderson localization problem always remains localized. As a result there is only a gapped Mott insulator phase in the theory. Comparisons with other studies and the parallel with disordered fermions in 2D are discussed. We conjecture that while for the physical cases N = 2 (XY) and N = 1 (Ising) the theory should have the ordered phase, it may not for N = 3 (Heisenberg).
We study the stability of the spin gap phase in the U (1) slave-boson theory of the t - J model in connection with the underdoped cuprates. We approach the spin gap phase from the superconducting state and consider the quantum phase transition of the slave bosons at zero temperature by introducing vortices in the boson superfluid. At finite temperatures, the properties of the bosons are different from those in the strange-metal phase and lead to modified gauge field fluctuations. As a result, the spin gap phase can be stabilized in the quantum-critical (QC) and quantum-disordered (QD) regime of the boson system. We also show that the regime of QD bosons with the paired fermions can be regarded as a strong-coupling version of the recently proposed nodal liquid theory.
The phase diagrams and phase transitions of bosons with short-ranged repulsive interactions moving in periodic and/or random external potentials at zero temperature are investigated with emphasis on the superfluid-insulator transition induced by varying a parameter such as the density. Bosons in periodic potentials (e.g., on a lattice) at T=0 exhibit two types of phases: a superfluid phase and Mott insulating phases characterized by integer (or commensurate) boson densities, by the existence of a gap for particle-hole excitations, and by zero compressibility. Generically, the superfluid onset transition in d dimensions from a Mott insulator to superfluidity is ``ideal,'' or mean field in character, but at special multicritical points with particle-hole symmetry it is in the universality class of the (d+1)-dimensional XY model. In the presence of disorder, a third, ``Bose glass'' phase exists. This phase is insulating because of the localization effects of the randomness and analogous to the Fermi glass phase of interacting fermions in a strongly disordered potential.The Bose glass phase is characterized by a finite compressibility, no gap, but an infinite superfluid susceptibility. In the presence of disorder the transition to superfluidity is argued to occur only from the Bose glass phase, and never directly from the Mott insulator. This zero-temperature superfluid-insulator transition is studied via generalizations of the Josephson scaling relation for the superfluid density at the ordinary \ensuremath{\lambda} transition, highlighting the crucial role of quantum fluctuations. The transition is found to have a dynamic critical exponent z exactly equal to d and correlation length and order-parameter correlation exponents \ensuremath{\nu} and \ensuremath{\eta} which satisfy the bounds \ensuremath{\nu}\ensuremath{\ge}2/d and \ensuremath{\eta}\ensuremath{\le}2-d, respectively. It is argued that the superfluid-insulator transition in the presence of disorder may have an upper critical dimension dc{d}_{c}dc​ which is infinite, but a perturbative renormalization-group calculation wherein the critical exponents have mean-field values for weak disorder above d=4 is also discussed. Many of these conclusions are verified by explicit calculations on a model of one-dimensional bosons in the presence of both random and periodic potentials. The general results are applied to experiments on 4He^{4}\mathrm{He}4He absorbed in porous media such as Vycor. Some measurable properties of the superfluid onset are predicted exactly [e.g., the exponent x relating the \ensuremath{\lambda} transition temperature to the zero-temperature superfluid density is found to be d/2(d-1)], while stringent bounds are placed on others. Analysis of preliminary data is consistent with these predictions.
We study superfluidity in strongly correlated systems by focusing three features as follows. First, interplay between superfluidity and other order parameters near quantum critical points is studied. The onset positions of the order parameters with or without superconductivity determines how they interact with each other. For the spin density wave (SDW) transition, we find that the quantum critical point shifts by order |∆|, where ∆ is pairing amplitude, so that the region of SDW order is smaller in the superconductor than in the metal. This shift is larger than the |∆|2 shift predicted by theories of competing orders which ignore Fermi surface effects. Second, we study a doped antiferromagnetic Mott insulator motivated by copper based high temperature superconductors. Based on spin and charge fractionalization, the algebraic charge liquid phase whose bosonic spinons have energy gap but fermionic holons form Fermi surfaces and its superconductivity are studied. We show fluctuations of local SDW order orientation lead to d-wave superconductivity, and its quantum critical pairing mechanisms are investigated. Also, a theory of bound states between spinons and holons is studied. We show that the fractionalization induces a doubling electron-like quasiparticles, which naturally induces the fractionalized Fermi liquid. The bound state theory describes exotic properties of the underdoped Cuprates such as “nodal-antinodal dichotomy” at a mean field level description. At last, we study superfluidity of fermions in optical lattices near the Feshbach resonance. Due to divergence of scattering length at the resonance, universality is observed and characteristics of superfluidity has been studied. We extend such universal properties by considering superfluid-insulator transition with an even integer number of fermions per unit cell. The resulting insulator interpolates between a band insulator of fermions and a Mott insulator of fermion pairs.
Two unrelated problems can be reduced to a model of a Bose gas interacting with a gauge field: (i) the effect of thermal fluctuations on a system of vortices in bulk superconductors in fields Hc1{\mathit{H}}_{\mathit{c}1}Hc1​\ensuremath{\ll}H\ensuremath{\ll}Hc2{\mathit{H}}_{\mathit{c}2}Hc2​, and (ii) charged, spinless excitations in two-dimensional (2D) strongly correlated electron systems. Both problems are important for the theory of high-temperature superconductors. We study this model in three regimes: at finite temperatures, assuming that the gauge field is purely transverse; at T=0, for the purely static (2D Coulomb) interaction; and at T=0, for a weak Coulomb interaction and a strong transverse one. Transverse interactions suppress the temperature of the superfluid transition significantly. A sufficiently strong transvese interaction is shown to produce a phase separation as the temperature decreases (in the absence of Coulomb repulsion). If there is Coulomb repulsion, the ground state does not have off-diagonal long-range order but the superfluid density is not zero unless the Coulomb constant exceeds a critical value. Sufficiently strong coupling to the transverse field destroys superfluidity as well. In the normal state formed at large couplings, the translational invariance is intact. We propose a bosonic ground state that is not superfluid at T=0. We discuss the implications of these results both for vortex liquids and strongly correlated electron systems.